Good Luck Charlie/WMG

The show is an alternate universe of Wizards of Waverly Place in which there is no magic

The kids don't fight and have switched roles. PJ, the oldest, is the dumb one. Teddy, the middle child, is the smart one. And Gabe, the youngest, is the mean one. Charlie is there because the parents weren't afraid to have another child because there's nothing like magic that would be a problem in the future.

All of the Credits Gags are Charlie's dreams

That would explain how each one is noticeably unrealistic compared to the show.

Again, the Rule of Funny comes into play to explain this.
Any gags with Charlie doing unrealistic stuff definitely qualifies as this. Some of those that don't contain Charlie may or may not be real.

Gabe was accidently switched at birth

In "Return to Super Fun Adventure Land", Gabe is in a commercial and has a fake mother. Upon seeing her, he points out his lack of resemblance to his real mother.

This is actually really possible. I mean, I realize it's fiction, but still, if Amy and Bob are both naturally blonde, it should be genetically impossible for them to have any brunet children. Three of their four children are blond. They are both blond. They should not be able to have any children with dark hair - the only possible hair colors for their offspring should be blond and maybe redhead.
- Unless he has a recessive gene, which means that his grandmother/father could have had brown hair in their youth, but that gene only decided to pop up now. Or his great grandparents had it, either or.
- Nix on that. The gene for blonde hair is the recessive gene. Brown hair is dominant. In theory, it is possible that there's a gene for brown hair in his family line somewhere, but it isn't possible that either Bob or Amy inherited that gene. (If they had, they would be brunettes.) Therefore, they only have recessive blond genes to give to Gabe, meaning that the above poster is correct: If Gabe really is their child, he should be as blond as the rest of the family.

The bad word that Charlie used in "Teddy on Ice" is "douchebag"

Consider the following:

It's something Amy called an offending motorist, so it's a pejorative, not merely an interjection.
The word was offensive enough to warrant Bob a bitch-slap from Mrs. Dabney.
P.J. knew the word was bad, but he didn't know what it meant. Being in high school, PJ has likely heard the F-bomb used in its literal sense, but it's quite likely that he knows Jack Squat about feminine hygiene.
And while P.J. is dumb, I'm sure he know what a**hole means, literally and derogatively.
- With all that said, I hope it was douchebag. The only other possible bad words that fit that criteria are cocksucker, and maybe sh_t-head, or f_ck-head. Nice job parenting, Amy. Especially if it was the first one of those three.
This troper is nearly positive that the word was c_nt. Evidence: the word was clearly an extremely offensive word based on the characters reactions to Charlie's use of it. Also, c_nt is a word which is meant to be offensive to women, which would explain why Mrs. Dabney was offended enough to slap Bob for using it.

Gabe is the illegitimate child of Teddy and Spencer; Teddy is several years older than anyone outside of the family thinks.

It sounds awfully far-fetched for a Disney Channel show, but consider the following:

Gabe looks nothing like either parent. Furthermore, he has no obvious resemblance to any of the older relatives seen on the show, including the maternal grandparents.
It would go a long way towards explaining Teddy's emotional attachment to Spencer. I know it's her "first love" and everything, but why would she be so gaga over someone she just went on a few dates with?
It would also explain Teddy's thinly-veiled contempt for Gabe. It kind of adds a weird subtext, doesn't it?
Have you ever noticed that Gabe and Spencer never seem to appear in the same scene? Almost as if Spencer was trying to avoid Gabe...
Also explains why Bob and Amy don't like Spencer. (Admittedly, they probably wouldn't like him either way, so this one is a bit of a stretch.)

Amy cheated on Bob in the past

There...I said it. She slept with another guy at some point in their marriage and Gabe was the result. This is the reason Bob freaks when he learns she dated another man after they broke up in high school and it's the reason Gabe's hair isn't blonde.
- This is admittedly the most obvious explanation, and certainly the simplest. However, this troper doubts that a Disney Channel show would allow such a thing to slip through, even unintentionally.
- Have you seen what slips through on Disney Channel? Go take a look at the Radar page for The Suite Life of Zack and Cody sometime...or Shake It Up...or even for this show. Some of it is pretty racy.

Amy dyes her hair.

She's actually a brunette, thus explaining why Gabe is a brunette.

Teddy's full name is Theodore Roosevelt Duncan.

Well, her initials ARE T.R. Duncan...it's possible that her parents thought she would be a boy and they wanted to name her after a president as badass as Teddy Roosevelt. Teddy is short for Theodore, too.